To reproduce:
1) Create new product
2) Switch platform to WebGL
3) Build and Run
4) First time it opens a localhost window normally and the project works
5) Build and Run again into same folder
6) It tries to open localhost, but it will say that "The page is unavailable"(message differs depending on browser)
6.5) Note: The build is made successfully. If you try to upload it to web and launch - it works perfectly
7) Build and Run again into different folder
8) It does not even try to open localhost, throwing this error message right at the end of building process:
InvalidOperationException: The process already finished.
System.Diagnostics.Process.Close (Int32 signo) <..>
8.5) Note: The build is made successfully. If you try to upload it to web and launch - it works perfectly

You can skip steps 5-6.5 and get identical results as listed

"Kind of workaround": Restart Unity editor. Go back to step 3.